Problem: Evaluate the following expression. Your answer must be exact. $\left(\dfrac{-5\sqrt{2}}{2}+\dfrac{5\sqrt{2}}{2}i\right)^2=$
The Strategy The easiest way to find $z^{n}$ for a complex number $z=({a}+{b}i)$ is using its modulus and argument. Therefore, our solution will consist of the following steps: Find the modulus and argument of $z$. [How is this done, in general?] Find the modulus and argument of $z^{n}$. [How is this done, in general?] Find the rectangular form $z^{n}$. Find the modulus and argument of $\left(\dfrac{-5\sqrt{2}}{2}+\dfrac{5\sqrt{2}}{2}i\right)$ $\left({\dfrac{-5\sqrt{2}}{2}}+{\dfrac{5\sqrt{2}}{2}}i\right)$ is of the form $({a}+{b}i)$, where ${a=\dfrac{-5\sqrt{2}}{2}}$ and ${b=\dfrac{5\sqrt{2}}{2}}$. Therefore: $\begin{aligned}r&=\sqrt{{a}^2 + {b}^2} \\\\&=\sqrt{ \left(-{\dfrac{5\sqrt{2}}{2}}\right)^2 + \left({\dfrac{5\sqrt{2}}{2}}\right)^2} \\\\&=\sqrt{{\dfrac{25}{2}}+{\dfrac{25}{2}}} \\\\&=5\end{aligned}$ Using the arctangent formula, we have: $\begin{aligned}\theta&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{\dfrac{5\sqrt{2}}{2}}}{{-\dfrac{5\sqrt{2}}{2}}}\right) \\\\&=-45^\circ\end{aligned}$ Since ${a=-{\dfrac{5\sqrt{2}}{2}}}$ is negative and ${b={\dfrac{5\sqrt{2}}{2}}}$ is positive, $\left(\dfrac{-5\sqrt{2}}{2}+\dfrac{5\sqrt{2}i}{2}\right)$ lies in Quadrant $2$. Therefore, $\theta$ must be between $90^\circ$ and $180^\circ$. Using the identity $\tan(180+\theta)=\tan(\theta)$, we know that the following is also a solution of the equation. $180^\circ+(-45^\circ)=135^\circ$ So $\theta = 135^{\circ}$. Find the modulus and argument of $\left(\dfrac{-5\sqrt{2}}{2}+\dfrac{5\sqrt{2}}{2}i\right)^2$ We found that the modulus and argument of $\left({\dfrac{-5\sqrt{2}}{2}}+{\dfrac{5\sqrt{2}}{2}}\right)$ are $5$ and $135^\circ$. Therefore, the modulus and argument of $\left({\dfrac{-5\sqrt{2}}{2}}+{\dfrac{5\sqrt{2}}{2}}\right)^2$ are $5^2=25$ and $(135^\circ)\cdot2=270^\circ$. Find the rectangular form of $\left(\dfrac{-5\sqrt{2}}{2}+\dfrac{5\sqrt{2}}{2}i\right)^2$ Since the argument is $270°$, we know the number lies on the negative side of the imaginary number axis and is therefore a negative pure imaginary number. Since the modulus is $25$, our solution is $-25i$. [What does this look like graphically?] [How do we find this algebraically?] Summary $\left(\dfrac{-5\sqrt{2}}{2}+\dfrac{5\sqrt{2}}{2}i\right)^2=-25i$